Finite Element Analysis of Composite Plateswith an Appli ation to the Paper Co klingProblemJuho Könnö a,∗, Rolf Stenberg a

aTKK, Helsinki University of Te hnology, Department of Mathemati s andSystems Analysis, PL1100, 02015 TKKAbstra tWe present a model for the laminated omposite plate problem based on the Reissner-Mindlin model and lassi al lamination theory. The weak problem based on thismodel is solved with the nite element method using stabilized MITC nite ele-ments. Detailed a priori estimates are given for the mixed nite element formulationof the problem. Expli it bounds for the ontinuity and oer ivity onstants of the oupling terms are presented, thus rening earlier results. Furthermore, we presentan appli ation of the theory to the paper o kling problem and investigate the ee tof boundary onditions on the small-s ale deformation of the solution numeri ally.Key words: Composite plates, MITC, nite element analysis, o kling, papermanufa turing∗ Corresponding author.Email addresses: juho.konnotkk.fi (Juho Könnö), rolf.stenbergtkk.fi(Rolf Stenberg).Preprint submitted to Finite Elements in Analysis and Design January 20, 2009

1 Introdu tionComposite stru tures are in reasingly ommon in industrial appli ations dueto their desirable properties, su h as ex eptionally high exural stiness-to-weight ratio and stru tural stability. In ne-grade paper manufa turing on-trolling the o kling of the paper sheet is of paramount importan e. Paper o kling o urs when a sheet of paper undergoes varying moisture onditions,e.g. in a drying pro ess during manufa turing. When studying paper o k-ling we are interested in the small-s ale dee tions of the paper sheet, whilstlarge-s ale deformations of the sheet are negle ted. The shape and size of the o kles depends heavily on the ber orientation, but also on the moisturegradient and the boundary onditions.To be able to inuen e the o klingbehavior during the manufa turing pro ess real-time simulations are needed,thus making e ient and reliable numeri al solution methods important.In this work, we present a model for a laminated omposite plate based on the lassi al lamination theory [18, whi h ouples the plate bending and planeelasti ity problems. We apply the laminated omposite model to modellingsheets of paper. A loser look is taken at an appli ation of the lassi al lam-ination theory to the paper o kling problem using a re ent material modelpresented in [12. Reissner-Mindlin kinemati assumptions are used for theplate model. We rst formulate the problem by means of minimization ofenergy. The orresponding weak equations are then dis retized by the niteelement method using stabilized MITC elements for the plate variables. Thein-plane displa ements are dis retized by standard rst order onforming niteelements.We extend the proof presented in [2 by giving expli it bounds for the on-2

tinuity and oer ivity onstants without assuming the plies to be of equalthi kness. Furthermore, we investigate the ee t of dierent physi ally rele-vant boundary onditions on the paper o kling problem, in whi h the quantityof interest is the lo al small-s ale dee tion of a paper sheet. Along with the ompli ated a tual physi al setup of a real-life paper o kling problem, thismakes it non-trivial to de ide the orre t boundary ondition for all the vari-ables involved, thus numeri al experiments are needed to verify the orre tsetup.2 The Reissner-Mindlin laminated plate modelThe plate bending problem is formulated for a moderately thi k plate of thi k-ness t in the domain Ω× (−t/2, t/2), where Ω ⊂ R

2. The kinemati unknownsare the transverse dee tion w, the rotation of the normals β and the in-plane displa ement u. The plate is subje ted to two types of loading, namelyin-plane loading f and transverse loading g. In omposite materials the on-ne tion between the planar elasti ity problem and the plate bending problem isintrodu ed dire tly through the onstitutive equations. In the following, bothdyadi and index notation for tensors are used in parallel. Indi es in Greekletters take the values 1, 2 and those in Roman letters the values 1, 2, 3.2.1 Constitutive relations for a single layerIn the following, all quantities with a tilde are always in the ply oordinate sys-tem, whereas those without are in the global oordinate system. The se ond-order transformation tensor Tij is dened by the rotation angle φ of the ply3

z

zk+1

zk−1

g f

Ω

z

y

xzk

Figure 1. Stru ture of the omposite plate oordinate system as follows

T =

cos φ − sin φ

sin φ cos φ

. (1)For a single layer, the onstitutive tensor, along with the stress and straintensors, must be transformed from the ply oordinate system to the global oordinate system. In index notation, the transformation for the onstitutivetensor is

Cijkl = TipTjqTkrTlsCpqrs. (2)For the stresses and strains it holdsσij = TipTjqσpq, ǫij = TipTjq ǫpq. (3)The onstitutive equation for a single ply in the ply oordinate system is

σ = C : ε. (4)4

The non-zero omponents of the onstitutive tensor in the ply oordinatesystem are dened by the six independent engineering parameters E1, E2, ν12,G12, G23, and G31 as [20, 10

C1111 = E1/(1 − ν12ν21), C2222 = E2/(1 − ν12ν21),

C1122 = ν12E2/(1 − ν12ν21), C1212 = G12,

C2323 = G23, C3131 = G31.

(5)2.2 Kinemati relationsWe denote by ε(u) := 1

2(∇u +∇uT ) the linear strain tensor. In the Reissner-Mindlin laminate model we allow a onstant shear deformation in the thi knessdire tion of the plate, and take into a ount the strain indu ed by the in-planedispla ements. The kinemati relations in the global oordinate system usingthe Reissner-Mindlin kinemati assumptions are [19

ǫαβ = ǫαβ(u) + zǫαβ(β), (6)ǫ3α =

∂w

∂xα

− βα, (7)ǫ33 = 0. (8)2.3 Constitutive relations for the plateHaving now formed the onstitutive relations for a single layer, we must formthe onstitutive tensor for the whole laminate. First, we form the membraneand shear stress resultants N and S, along with the bending moment resultant5

M , by integrating over the thi kness of the laminate. Ck is the onstitutivetensor for a single ply. The stru ture of the laminate is depi ted in Figure 1.With this notation the for e resultants for a laminate with n layers readNαβ =

∫ t/2

−t/2

σαβdz =n

∑

k=1

∫ zk

zk−1

σαβdz, (9)Mαβ =

∫ t/2

−t/2

σαβzdz =n

∑

k=1

∫ zk

zk−1

σαβzdz, (10)Sα =

∫ t/2

−t/2

σ3αdz =n

∑

k=1

∫ zk

zk−1

σ3αdz. (11)We also assume individual layers to be homogeneous in the thi kness dire tion.Inserting the kinemati relations (6)-(8) we an write the stress and momentresultants with the help of the following se ond- and fourth-order tensors [10

Aαβγδ =n

∑

k=1

∫ zk

zk−1

Cαβγδdz =n

∑

k=1

(zk − zk−1)Ckαβγδ, (12)

Bαβγδ =n

∑

k=1

∫ zk

zk−1

Cαβγδzdz =1

2

n∑

k=1

(z2

k − z2

k−1)Ck

αβγδ, (13)Dαβγδ =

n∑

k=1

∫ zk

zk−1

Cαβγδz2dz =

1

3

n∑

k=1

(z3

k − z3

k−1)Ckαβγδ, (14)

A∗

αβ =n

∑

k=1

∫ zk

zk−1

C3α3βdz =n

∑

k=1

(zk − zk−1)Ck3α3β . (15)With these denitions the resultants an be written as

N = A : ε(u) + B : ε(β), (16)M = B : ε(u) + D : ε(β), (17)S = A∗· (∇w − β). (18)6

2.4 The Reissner-Mindlin plate modelThe energy asso iated with ea h deformation mode an be omputed by rstmultiplying ea h strain omponent with the orresponding stress resultant,and then integrating over the domain Ω o upied by the plate. We introdu ethe following s aling for the variables, material tensors and loadings:

u → 1

tu,

β → β,

w → w

,

A → 1

tA,

B → 1

t2B,

D → 1

t3D,

A∗ → 1

tA∗

,

f → 1

t2f ,

g → 1

t3g

. (19)By s aling the unknowns as above, the total energy is s aled by a fa tor of t3.The s aled total energy of the plate an be written as [2

Π(u, w, β) =1

2

∫

Ω

ε(u) : A : ε(u)dΩ

+∫

Ω

ε(u) : B : ε(β)dΩ

+1

2

∫

Ω

ε(β) : D : ε(β)dΩ (20)+

1

2t2

∫

Ω

(∇w − β)·A∗· (∇w − β)dΩ

−∫

Ω

f ·udΩ−∫

Ω

gwdΩ.The s aled shear for e q isq = t−2A∗· (∇w − β). (21)The shear for e plays a key role in the error analysis of the asso iated niteelement method. In the analysis we treat the shear for e as an independent7

unknown thus arriving at a mixed formulation for the original problem.2.5 Weak form of the equationsTo derive the asso iated weak form of the problem we minimize the energyexpression (20) with respe t to the kinemati variables. The problem is: Find(u, w, β) ∈ U × W × V su h, that

(A : ε(u), ε(v)) + (B : ε(v), ε(β)) = (f , v) ∀v ∈ Uand(B : ε(u), ε(η)) + (D : ε(β), ε(η))

+ t−2(A∗· (∇w − β), (∇ω − η)) = (g, ω), ∀(ω, η) ∈ W × V .The minimization results in two equations, a standard plane elasti ity problemand a Reissner-Mindlin plate problem. The two problems are oupled by anadditional term onne ting the rotational and the in-plane degrees of freedomthrough the tensor B. U ⊂ [H1(Ω)]2, W ⊂ H1(Ω) and V ⊂ [H1(Ω)]2 aresuitably hosen variational spa es, see e.g. [15, 7.2.6 Coer ivity resultWe prove a oer ivity result similar to the one presented in [2. We extend theproof by not assuming the plate to be evenly distributed into plies of thi knesstn, and present expli it relations for the oer ivity and ontinuity onstants. Asexpe ted, the upper bound remains independent of the thi kness onguration8

of the laminate. The lower bound, on the other hand, depends both on thenumber of layers and the thi kness ratio of the individual plies. The theoremshows, that in the worst- ase s enario a laminate with a large number of verythin plies leads to rapid deterioration of the oer ivity onstant. However, ifthe laminate ontains both thi k and thin plies, the thi k plies dominate thelower bound and the onstant does not deteriorate. In this sense, the estimategiven in the theorem is somewhat pessimisti but robust, sin e it takes intoa ount also the worst possible ply onguration.Theorem 1 There exists positive onstants C1, C2 su h that for every pair ofsymmetri tensors τ , σ ∈ [L2(Ω)]4 it holdsC1(‖τ‖2

0+ ‖σ‖2

0) ≤ (A : τ , τ ) + 2(B : τ , σ) + (D : σ, σ) ≤ C2(‖τ‖2

0+ ‖σ‖2

0),(22)in whi h A, B, D are the tensors dened above, and it is assumed that the onstitutive tensor C ∈ [L2(Ω)]4×4 and is symmetri positive denite. Fur-thermore, denoting by δ the ratio of the thi kness of the thinnest layer to thethi kness of the whole laminate, δ = 1

tmink hk, we have C1 ∼ nδ3 and C2 ∼ 1.Proof:Writing the expression as a formal matrix produ t and taking into a - ount that B is symmetri we have

(A : τ , τ ) + 2(B : τ , σ) + (D : σ, σ) =

τ

σ

,n

∑

k=1

Ak Bk

Bk Dk

τ

σ

. (23)Thus the theorem is true, if the oe ient matrix is bounded and positivedenite. Turning our attention to the matrix for a single ply, we use deni-9

tions (12)(14) to write the orresponding matrix in the form

Ak Bk

Bk Dk

=

1

t(zk − zk−1)

1

2t2(z2

k − z2

k−1)

1

2t2(z2

k − z2

k−1) 1

3t3(z3

k − z3

k−1)

Ck 0

0 Ck

(24)Sin e the last matrix is blo k diagonal, the produ t of these two matri es ommutes. Thus, it is su ient to prove the positive deniteness property forboth matri es separately.The tensor Ck is symmetri and positive denite by denition, hen e we onlyneed to know the eigenvalues of the 2 × 2 matrix

Rk :=

1

t(zk − zk−1)

1

2t2(z2

k − z2k−1)

1

2t2(z2

k − z2k−1)

1

3t3(z3

k − z3k−1)

. (25)The eigenvalues of a Rk are [8

λ1(Rk) =tr(Rk)

2

1 −√

√

√

√1 − 4 det(Rk)

tr(Rk)2

,

λ2(Rk) =tr(Rk)

2

1 +

√

√

√

√1 − 4 det(Rk)

tr(Rk)2

.For the matrix Rk the determinant and the tra e an be written asdet(Rk) =

1

3t4(z3

k − z3

k−1)(zk − zk−1) −1

4t4(z2

k − z2

k−1)2 =

1

12t4(zk − zk−1)

4,

tr(Rk) =1

t(zk − zk−1) +

1

3t3(z3

k − z3

k−1).10

Denoting the thi kness of the k:th ply by hk = zk−zk−1 the determinant readsdet(Rk) =

h4k

12t4> 0,and for the tra e it holds

tr(Rk) ≥1

t(zk − zk−1) =

hk

t> 0.The eigenvalues are real and positive sin e

0 <4 det(Rk)

tr(Rk)2≤ h2

k

3t2< 1.For every k it holds zk ∈ [−t/2, t/2]. Thus we an estimate the tra e fromabove by

tr(Rk) =1

t(zk − zk−1) +

1

3((

zk

t)3 − (

zk−1

t)3) ≤ 2

t(zk − zk−1) =

2hk

t.

Using the inequality 1 −√

1 − x ≥ x2, x ∈ [0, 1], we have for the smallesteigenvalue

λ1(Rk) ≥tr(Rk)

2

2 det(Rk)

tr(Rk)2=

h3

k

24t3.By Weyl's inequality [8, Theorem 4.3.1

λ1(n

∑

k=0

Rk) ≥n

∑

k=0

λ1(Rk) =1

24t3

n∑

k=0

h3

k ≥ nδ3

24. (26)For the largest eigenvalue we have

λ2(n

∑

k=0

Rk) ≤n

∑

k=0

λ2(Rk) ≤n

∑

k=0

tr(Rk) =1

t(zn − z0) +

1

3t3(z3

n − z3

0) =

13

12. (27)11

Thus the matrix is uniformly bounded from above, and the onstant C2 is ompletely independent of the ply onguration of the laminate. The lowerbound, on the other hand, is related to the thi kness ratio δ as C1 ∼ nδ3. 2

3 The nite element modelIn the nite element formulation we use the stabilized version of the well-known MITC elements of the rst order originally presented in [3, 4 andlater in [6, 7, 16. In the MITC family of elements, the lo king phenomenonis avoided by proje ting the rotation β onto a suitable subspa e of H(rot, Ω)using the redu tion operator Rh dened below. Using linear MITC elementsrequires a modi ation of the energy expression to a hieve a stable method.The stabilization is obtained by lo ally repla ing the fa tor t−2 by (t+αhK)−2in the shear energy expression [7, 16, in whi h hK is the lo al mesh param-eter and α a suitably hosen stabilization parameter, typi ally α = 0.1. Forhigher order elements additional terms are required in the bilinear form forthe stabilization when using equal order interpolation. However, an equivalentway is to use additional inner degrees of freedom for the rotation and thenon-modied bilinear form [14. For simpli ity, we present the element spa esand onvergen e results only for the rst-order elements. For details on higherorder elements, see [15, 6, 7. 12

3.1 ImplementationWe denote by Kh a shape-regular quadrilateral mesh over the domain Ω. Thenite element spa es for quadrilateral rst-order elements areVh = η ∈ V | η|K ∈ [Q1(K)]2, ∀K ∈ Kh,

Wh = w ∈ W | w|K ∈ Q1(K), ∀K ∈ Kh,

Uh = u ∈ U | u|K ∈ [Q1(K)]2, ∀K ∈ Kh.The dis rete spa e Γh for the shear for e is dened on the referen e elementK asΓh(K) = s = (s1, s2)| s1 = a + bx2, s2 = c + dx1, for some a, b, c, d ∈ R.We dene for ea h element K ∈ Kh the shear spa e asΓh(K) = s = (s1, s2)| s(x, y) = J−T

K s(F−1

K (x, y)), for some s ∈ Γh(K).Here FK is the mapping from the referen e element K to the element K, andJK the orresponding Ja obian matrix. This allows us to dene the redu tionoperator Rh : Vh → Γh(K) from the following onditions on the edges E ofea h element

∫

E((Rhs − s)· τ ) = 0.Here τ is the tangent to ea h edge E of the element K.For the redu tion operator it holds Rh∇w = ∇w [7. Thus the dis rete for-mulation for the problem is: Find (uh, wh, βh) ∈ Uh × Wh × Vh su h that it13

holds(A : ε(uh), ε(vh)) + (B : ε(vh), ε(βh)) = (f , vh) ∀vh ∈ Uhand

(B : ε(uh), ε(ηh)) + (D : ε(βh), ε(ηh))

+∑

K∈K

1

t2 + αh2K

(A∗· (∇wh − Rhβh), (∇ωh − Rhηh)) = (g, ωh)

∀(ωh, ηh) ∈ Wh × Vh.3.2 Theoreti al onsiderationsGenerally speaking, all results for the standard Reissner-Mindlin plate an beextended to the ase of a laminated plate. The norm equivalen e in Theorem 1is essential in proving the onvergen e properties. It shows that the oupledequation is ellipti with respe t to all the kinemati variables involved and theupper bound does not depend on the ply stru ture of the laminate. Thus we an show that the laminated Reissner-Mindlin plate model has the same desir-able onvergen e properties as the homogeneous plate [2, 11, 16 by onsideringboth parts of the equation separately and using Theorem 1. The regularityestimates needed in the analysis an be found in [15, 1, 11. For the platemodel not involving the in-plane displa ements we use a stability- onsisten ytype argument. The stability property only holds in a spe i mesh depen-dent norm, see [14, 16. Furthermore, using a mesh-stabilized method indu esa onsisten y error due to the stabilizing term in the dis rete bilinear form.The onsisten y error an be bouded from above by the weighted norm of theshear for e q. For the plane elasti ity problem the estimates follow from the14

lassi al theory of nite element methods [5.The main onvergen e result gives us onvergen e relative to h in the domainΩ, where h is the global mesh parameter. More pre isely, one has the followinga priori error estimate for the oupled problem with a su iently smoothsolution

‖β − βh‖1 + ‖w − wh‖1 + t‖q − qh‖0 + ‖q − qh‖−1 + ‖u − uh‖1 ≤ Ch.Here the onstant C depends only on the loading.4 Modelling paper o kling with the plate modelIn general, paper is a very inhomogeneous material with arbritrary ber dire -tions and strong anisotropies. However, a valid simpli ation is to onsider asheet of paper as a material omposed of layers with individual ber dire tionpatterns, dened by the angle Θ = π

2− φ in ea h individual data blo k. These ond lo al material parameter needed to des ribe the material is the level oflo al anisotropy denoted by ξ. In addition, only the easily measurable globalYoung's modulus E and poisson number ν are needed. The denitions of thelo al quantities Θ and ξ are presented in Figure 2.The level of anisotropy ξ is dened as the ratio of the largest ber on en-tration to the on entration in the dire tion perpendi ular to it. This an beinterpreted geometri ally as the ratio of the pri ipal axes of the ellipsoid inFigure 2, namely ξ = a/b. Some pra ti al methods for measuring the angleand anisotropy data from a paper sample are presented in [12, 13. The thirdlo al parameter needed for the model is the moisture per entage β, whi h is15

n2

b

a

n1MD

CD

Θ

Figure 2. Denition of the orientation angle Θ and anisotropy ξ. MD is the ma hinedire tion and CD the ross-ma hine dire tion. The n1 axis points in the dire tion ofstrongest anisotropy, and n2 is perpendi ular to this dire tion.allowed to vary not only in the data blo ks, but also layerwise in the thi knessdire tion, whi h is mostly the ase in the a tual modelled situations. The fol-lowing model for the linear and elasti o kling of paper is presented originallyin [12.4.1 Young's modulus and Poisson ratioIn pra ti e, only the global Young's moduli EMD and ECD are usually mea-sured from a paper sample. The lo al Young's moduli in the ply oordinatesystem are derived from these two quantities, assuming they depend only onthe moisture level β and anisotropy ξ. By geometri inspe tion and the de-nition of ξ, it evidently holds

E1

E2

= ξ. (28)Sin e the geometri mean E =√

E1E2 of the Young's moduli remains approx-imately onstant regardless of the orientation, we an write the lo al moduli16

in the following form treating the geometri mean as an invariantE1 = E

√

ξ, E2 =E√ξ. (29)Empiri al results show [12, 9, that for most paper qualities the moisturedependen e of the geometri average E is linear. In this paper we use thedependen e presented in [12,

E = (−0.25β + 6.5) GPa. (30)The Poisson ratios are dened from the global quantities with the help of thelo al Young's moduli. By Maxwell's re ipro al relation it holdsν12

ν21

=E1

E2

= ξ. (31)From this we have analogously to the previous aseν12 = ν

√

ξ, ν21 =ν√ξ. (32)On e again, experiments support the fa t that the global Poisson numberdepends linearly on the moisture ontent for the majority of paper qualities,as noted in [12 and the referen es therein. Here the following dependen e isused

µ = 0.015β + 0.150. (33)The shear modulus is nally dened analoguously to isotropi materials [12,11 from the geometri averages E and ν. Thus we get the following expression17

not depending on the lo al anisotropy ξ:G12 =

E

2(1 + µ)=

6.5 − 0.25β

2.3 + 0.03βGPa. (34)The model used here is also onsistent with the isotropi ase, sin e for ho-mogeneous material the anisotropy ξ = 1. In this ase we have E1 = E2 = E,and ν12 = ν21 = ν, as expe ted. For G13 and G23 we use the referen e valuesfrom [12. For a more in-depth analysis and validation of the material model, f. [12 and the referen es therein.4.2 Moisture loadingThe primary sour e of loading during the drying pro ess is the shrinking ofthe paper with dropping moisture level. The shrinking phenomenon an bemodelled mathemati ally exa tly in the same way as heat expansion, with themoisture expansion oe ients depending on the lo al anisotropy ξ as follows:

α1(ξ) = 0.0006 − 0.00015√

ξ − 1, (35)α2(ξ) = 0.0006 +

√

7 × 10−8(ξ − 1). (36)These are values for a typi al paper sample taken from [12. The orrespondingstrain is obtained from the oe ients asǫmoisture

ij =

α1(ξ)β 0

0 α2(ξ)β

. (37)As with heat expansion, the shrinking only indu es strains in the prin ipal18

ply oordinate dire tions. However, globally we get also shear deformations asthe strains and orresponding stresses are transferred from the ply oordinatesystem to the global oordinate system. The moisture ontent β is allowedto vary from layer to layer. Moreover, this auses no mathemati al problemssin e the indu ed stress is omputed layerwise.4.3 Boundary onditionsFor the in-plane displa ement u assigning the boundary onditions is straigh-forward. The Reissner-Mindlin plate model, on the other hand, a ounts fora variety of dierent boundary onditions. We denote by βn and βτ the ro-tations in the normal and tangent dire tions of the boundary, respe tively.On ea h boundary we an set one of the following physi ally sound boundary onditions for the plate variables (w, β) [19:- Clamped, xing w and both omponents of β- Soft lamped, xing w and βn- Simply supported, xing only w- Hard simply supported, xing w and βτ- Free, no restri tions on the variablesThe a tual boundary onditions in the paper ma hine during the manufa tur-ing pro ess are problemati to model a urately. Sin e no large s ale urvingusually o urs during the pro ess, the natural hoi e is to set w = 0 on alledges. We take a loser look at hoosing the orre t boundary onditions forthe plate and in-plane variables in the numeri al experiments below.19

5 Numeri al testsIn the numeri al experiments, we on entrate on the ee t of dierent bound-ary onditions on the o kling behaviour of the paper sheet. All tests wereperformed on a 192× 192 square mesh with rst order stabilized MITC4 [16elements. In the results, we lter out the lowest 20 per ent of wavelengths. Arelatively low-order high-pass lter based on Fourier transform te hniques isused. Filtering is used to fo us on the lo al shape of the o kles instead of thelarge-s ale deformations of the paper sheet. The size of the simulated papersample is 1000 mm× 1000 mm in all of the omputations. All results are pre-sented in millimetres. The omputations were performed with the ommer ialNumerrin software [17, in whi h the elements presented are implemented.5.1 Ee t of dierent boundary onditionsIn the numeri al experiments we onsider the two most important bound-ary onditions for the Reissner-Mindlin model, namely simply supported and lamped boundaries. Furthermore, we have additional boundary onditionsfor the in-plane displa ements. Here we onsider either un onstrained or om-pletely xed planar displa ements. The aim of these numeri al experiments isto nd out whi h boundary onditions ae t the o kling behavior the most,and thus the hoi e of whi h plays the most prominent role in the modellingpro ess. The resulting o kling behavior for these four ases is presented inFigures 3 and 4.As is evident from the results, dierent boundary onditions for the planardispla ements give ompletely dierent o kling behaviour. With the un on-20

−4

−3

−2

−1

0

1

2

3

4

5

−4

−3

−2

−1

0

1

2

3

4

5

Figure 3. Co kling behavior with un onstrained planar displa ements, top view ofthe 1000 mm× 1000 mm sample. Dieren e between bla k and white is 10 mm. Onthe left we have simply supported boundary onditions and on the right lampedboundary onditions.

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Figure 4. Co kling behavior with onstrained planar displa ements, top view of the1000 mm × 1000 mm sample. Dieren e between bla k and white is 0.8 mm. Onthe left we have simply supported boundary onditions and on the right lampedboundary onditions.strained boundary onditions the amplitude of the o kling is roughly a de adelarger than for the onstrained ase. Thus hoosing the boundary onditionfor the planar dipla ements seems to play the most ru ial role. This an beexplained by onsidering the moisture expansion behaviour of the laminate.Sin e the in-plane displa ements u and rotations β are oupled, xing thein-plane displa ements indu es moment loadings on the laminate a ounting21

for larger and smoother total bending of the plate.Next we onsider the hoi e of boundary ondition for the plate variables wand β. From Figure 5 we an see, that large values of absolute dieren ebetween the solutions are on entrated near the boundaries, whereas in therest of the domain the dieren e is rather smooth. This suggest that the hoi eof the boundary ondition for the plate equation has less signigan e than the hoi e for the in-plane displa ements, sin e the overall lo al o kling behaviourand the amplitude of the o kles is virtually un hanged.

0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 5. The absolute dieren e between the simply supported and lamped solu-tion for un onstrained planar displa ements on the left, and for the onstrained aseon the right. Top view of the 1000 mm×1000 mm sample. Dieren e between bla kand white is 5 mm on the left and 0.5 mm on the right.6 Con luding remarksWe noti ed that the laminate model based on lassi al lamination theory andReissner-Mindlin kinemati assumptions is well-dened and inherits the on-vergen e properties from those of plane elasti ity and Reissner-Mindlin platetheory. We also provided a urate bounds for the oer ivity and ontinuity onstants of the oupled problem, and pointed out the worst- ase s enario in22

whi h theory suggest problems may arise. The model appears to be quite well-suited to modelling the paper o kling phenomenon using the material modelpresented in [12 and allows relatively straightforward implementation of moreadvan ed material models as proposed in [13. We were also able to determinethe signigan e of dierent boundary onditions on the o kling phenomenon,and identied the hoi e of boundary ondition for the in-plane displa ementsto be the ru ial fa tor in the model when fo using on the small-s ale o kling.7 A knowledgementsThe rst author has been supported by a grant from the Finnish CulturalFoundation. This work has been supported by TEKES, The National Te h-nology Agen y of Finland (proje t KOMASI, de ision number 40322/07).

23

Referen es[1 Arnold, D. N., Falk, R. S., 1989. A uniformly a urate nite elementmethod for the Reissner-Mindlin plate. SIAM Journal on Numeri al Anal-ysis 26, 12761290.[2 Auri hio, F., Lovadina, C., Sa o, E., 2001. Analysis of mixed niteelements for laminated omposite plates. Comp. Methods Appl. Me h.Engrg 190, 47674783.[3 Bathe, K.-J., Dvorkin, E., 1985. A Four-Node Plate Bending ElementBased on Mindlin/Reissner Plate Theory and a Mixed Interpolation. In-ternational Journal for Numeri al Methods in Engineering, 21, 367383.[4 Bathe, K.-J., Dvorkin, E. N., 1986. A Formulation of General ShellElements-The Use of Mixed Interpolation of Tensorial Components. Int.Journal for Numeri al Methods in Engineering, 22, 697722.[5 Braess, D., 2001. Finite Elements. Cambridge University Press.[6 Brezzi, F., Bathe, K.-J., Fortin, M., 1989. Mixed-interpolated elementsfor Reissner-Mindlin plates. Internat. J. Numer. Methods Engrg. 28 (8),17871801.[7 Brezzi, F., Fortin, M., Stenberg, R., 1991. Error analysis of mixed-interpolated elements for Reissner-Mindlin plates. Mathemati al Modelsand Methods in Applied S ien es 2, 125151.[8 Horn, R. A., Johnson, C. R., 1985. Matrix Analysis. Cambridge UniversityPress.[9 Htun, M., Fellers, C., 1982. The invariant me hani al properties of ori-ented handsheets. TAPPI journal 65, 113117.[10 Kere, P., Lyly, M., 2008. Reissner-Mindlin-Von Kármán type plate modelfor nonlinear analysis of laminated omposite stru tures. Computers and24

Stru tures 86 (9), 10061013.[11 Könnö, J., 2007. Finite element analysis of omposite laminates (inFinnish). Master's thesis, Helsinki University of Te hnology.URL http://math.tkk.fi/u/jkonno/dtfinal.pdf[12 Leppänen, T., 2007. Ee t of ber orientation on o kling of paper. Ph.D.thesis, University of Kuopio.[13 Lipponen, P., Leppänen, T., Kouko, J., Hämäläinen, J., 2008. Elasto-plasti approa h for paper o kling phenomenon: On the importan eof moisture gradient. International Journal of Solids and Stru tures 45,35963609.[14 Lyly, M., 2000. On the onne tion between some linear triangularReissner-Mindlin plate bending elements. Numer. Math. 85 (1), 77107.[15 Lyly, M., Niiranen, J., Stenberg, R., 2006. A rened error analysis ofMITC plate elements. Mathemati al Models and Methods in AppliedS ien es 16 (7), 967977.[16 Lyly, M., Stenberg, R., Vihinen, T., 1993. A stable bilinear element forthe Reissner-Mindlin plate model. Comput. Methods Appl. Me h. Engrg.110 (3-4), 343357.[17 Numerola Oy, 2008. http://www.numerola.fi.[18 Reddy, J., 2004. Me hani s of Laminated Composite Plates and Shells.CRC Press.[19 Timoshenko, S. P., Woinowsky-Krieger, S., 1959. Theory of Plates andShells. M Graw-Hill.[20 Vinson, J. R., 1999. The Behavior of Sandwhi h Stru tures of Isotropi and Composite Materials. Te hnomi .25